number of prime ideals in a number field

Proof. Let 𝒪 be the ring of integers of a number field. If p is a rational prime number, then the principal ideal(p) of 𝒪 does not coincide with (1)=𝒪 and thus (p) has a set of prime ideals of 𝒪 as factors. Two different (positive) rational primes p and q satisfy

gcd⁡((p),(q))=(p,q)=(1),

since there exist integersx and y such that x⁢p+y⁢q=1 and consequently 1∈(p,q). Therefore, the principal ideals (p) and (q) of 𝒪 have no common prime ideal factors. Because there are http://planetmath.org/node/3036infinitely many rational prime numbers, also the corresponding principal ideals have infinitely many different prime ideal factors.